Set Theory Talks

Eric Pacuit, University of Maryland

Maximal almost disjoint (MAD) families and their relatives have been an important area of combinatorial and descriptive set theory since at least the 60s. In this talk I will discuss some relatives of MAD families, focussing on eventually different families of functions f:ω→ωf:ω→ω and eventually different sets of permutations p∈S(ω)p∈S(ω). In the context of MAD families it has been fruitful to consider various strengthenings of the maximality condition to obtain several flavors of 'strongly' MAD families. One such strengthening that has proved useful in recent literature is that of tightness. Tight MAD families are Cohen indestructible and come with a properness preservation theorem making them nice to work with in iterated forcing contexts.

I will introduce a version of tightness for maximal eventually different families of functions f:ω→ωf:ω→ω and maximal eventually different families of permutations p∈S(ω)p∈S(ω) respectively. These tight eventually different families share a lot of the nice, forcing theoretic properties of tight MAD families. Using them, I will construct explicit witnesses to ae=ap=ℵ1ae=ap=ℵ1 in many known models of set theory where this equality was either not known or only known by less constructive means. Working over LL we can moreover have the witnesses be Π11Π11 which is optimal for objects of size ℵ1ℵ1 in models where CHCH fails. These results simultaneously strengthen several known results on the existence of definable maximal sets of reals which are indestructible for various definable forcing notions. This is joint work with Vera Fischer.

Dave Marker, University of Illinois at Chicago
Real closures of ω1ω1-like models of PA

D'Aquino, Knight and Starchenko showed the real closure of a model of Peano Arithmetic is recursively saturated. Thus any two countable models of PA with the same standard system have isomorphic real closures. Charlie Steinhorn, Jim Schmerl and I showed that even for ω1ω1-like model of PA the situation is very different. We construct 2ℵ12ℵ1 recursively saturated elementarily equivalent ω1ω1-like models of PA with the same standard system and non-isomorphic real closures.

Paradoxes of perfectly small sets

We define a set of real numbers to be perfectly small if it has perfectly many disjoint translates. Such sets have a strong intuitive claim to being probabilistically negligible, yet no non-trivial measure assigns them all a value of 0. We will prove from a moderate amount of choice that any total extension of Lebesgue measure concentrates on a perfectly small set, suggesting that for any such measure, translation-invariance fails 'as badly as possible.' From the ideas of this proof, we will also derive analogues of well-known paradoxes of randomness, specifically Freiling's symmetry paradox and the infinite prisoner hat puzzle, in terms of perfectly small sets. Finally, we discuss how these results constrain what a paradox-free set theory can look like and some related open questions.

Induction and collection up to definable elements: calibrating the strength of parameter-free ΔnΔn-minimization.

In this talk we shall deal with fragments of first-order Peano Arithmetic obtained by restricting the conclusion of the induction or the collection axiom to elements in a prescribed subclass DD of the universe. Fix n>0n>0. The schemes of ΣnΣn-induction up to ΣmΣm-definable elements and the schemes of ΣnΣn-collection up to ΣmΣm-definable elements form two families of subtheories of IΣnIΣn and BΣnBΣn, respectively, obtained in this way.

The properties of ΣnΣn-induction up to ΣmΣm-definable elements for n≥mn≥m are reasonably well understood and interesting applications of these fragments are known. However, an analysis of the case n<mn<m was pending. In the first part of this talk, we address this problem and show that it is related to the following general question: 'Under which conditions on a model MM can we prove that every non-empty ΣmΣm-definable subset of MM contains some ΣmΣm-definable element?'

In the second part of the talk, we show that, for each n≥1n≥1, the scheme of ΣnΣn-collection up to ΣnΣn-definable elements provides us with an axiomatization of the Σn+1Σn+1-consequences of BΣnBΣn. As an application, we obtain that BΣnBΣn is Σn+1Σn+1-conservative over parameter-free ΔnΔn-minimization (plus IΣn−1IΣn−1), thus partially answering a question of R. Kaye.

This is joint work with F.Félix Lara-Martín (University of Seville).

Andrés Villaveces, Universidad Nacional de Colombia – Bogotá
Two logics, and their connections with large cardinals / Questions for BDGM: Part II

In the past couple of years I have been involved (joint work with Väänänen and independently with Shelah) with some logics in the vicinity of Shelah's L1κLκ1 (a logic from 2012 that has Interpolation and a very weak notion of compactness, namely Strong Undefinability of Well-Orderings, and in some cases has a Lindström-type theorem for those two properties). Our work with Väänänen weakens the logic but keeps several properties. Our work with Shelah explores the connection with definability of AECs.

These logics seem to have additional interesting properties under the further assumption of strong compactness of a cardinal, and this brings them close to recent work of Boney, Dimopoulos, Gitman and Magidor [BDGM].

During the first lecture, I plan to describe two games and a syntax of two logics: Shelah's L1κLκ1 and my own logic (joint work with Väänänen) L1,cκLκ1,c. I will stress some of the properties of these logics, without any use of large cardinal assumptions.

During the second lecture, I plan to enter rather uncharted territory. I will describe some constructions done by Shelah (mostly) under the assumption of strong compactness, but I also plan to bring these logics to a territory closer to the work of [BDGM]. This second lecture will have more conjectures, ideas, and (hopefully interesting) discussions with some of the authors of that paper.

Dave Marker, University of Illinois at Chicago
Real closures of ω1ω1-like models of PA

D'Aquino, Knight and Starchenko showed the real closure of a model of Peano Arithmetic is recursively saturated. Thus any two countable models of PA with the same standard system have isomorphic real closures. Charlie Steinhorn, Jim Schmerl and I showed that even for ω1ω1-like model of PA the situation is very different. We construct 2ℵ12ℵ1 recursively saturated elementarily equivalent ω1ω1-like models of PA with the same standard system and non-isomorphic real closures.

Roman Kossak, CUNY
Automorphisms, Jónsson Models, and Satisfaction Classes

25 years ago I wrote a paper on four open problems concerning recursively saturated models of PA. The problems are still open. I will talk about two of them: (1) Let M be a countable recursively saturated model of PA. Can every automorphism of M be extended to some recursively saturated elementary end extension of M? (2) Is there a recursively saturated model of PA that has no recursively saturated elementary submodel of the same cardinality as the model? I will present some partial results involving partial inductive satisfaction classes.